Operator Theory on the Pentablock

TL;DR

This paper explores the operator theory on the pentablock, introducing new classifications of isometries, a Wold-type decomposition, and models for pure isometries, advancing understanding of this complex domain.

Contribution

It provides an algebraic characterization of pentablock isometries, introduces a new component called the quasi-pentablock unitary, and develops functional models and invariant subspace representations.

Findings

Characterization of pentablock unitaries and isometries

Introduction of the quasi-pentablock unitary component

Development of a Beurling-Lax-Halmos type model

Abstract

The pentablock, denoted as $\cP,$ is defined as follows: $$\cP= \left\{ (a_{21}, {\rm tr}(A), {\rm det}(A)) : A = [a_{ij}]_{2 \times 2} \text{ with } \|A\|<1 \right\}.$$ It originated from the work of Agler--Lykova--Young in connection with a particular case of the $\mu$-synthesis problem. It is a non-convex, polynomially convex, $\mathbb{C}$-convex, star-like about the origin, and inhomogeneous domain. This paper deals with operator theory on the pentablock. We study pentablock unitaries and isometries, providing an algebraic characterization of pentablock isometries. En route, we provide the Wold-type decomposition for pentablock isometries, which consists of three parts: the unitary part, the pure part, and a new component. We define this novel component as the quasi-pentablock unitary and provide a functional model for it. Additionally, a model for a class of pure pentablock isometries has been found, along with some examples. Furthermore, a representation resembling the Beurling-Lax-Halmos paradigm has been presented for the invariant subspaces of pentablock pure isometries.